Quantum criticality perspective on population fluctuations of a localized electron level

1. Quantum criticality perspective on population fluctuations of a localized electron level Vyacheslavs (Slava) Kashcheyevs Collaboration: Christoph Karrasch, Volker Meden (RTWH AachenU., Germany) Theresa Hecht, Andreas Weichselbaum (LMU Munich, Germany) AvrahamSchiller (Hebrew U., Jerusalem,Israel) “The Science of Complexity”, Minerva conference, Eilat, March31st, 2009

2. Quantum criticality perspective on population fluctuations of a localized electron level

3. Average population <n–> V– Ω 1 0 Start non-interacting ε– EF – Increase level energy ε– Critical ε*= EF

4. V– Ω 1 b 0 Add on-site interactions Average population <n–> ε– EF – + U V+ ε+ Increase level energy ε– Without V_, b: Two disconnected, orthogonal ground states, “critical” at ε–=ε*

5. V– Ω 1 b 0 Results in a nutshell Average population <n–> “narrow” “broad” ε– EF – + U V+ ε+ Increase level energy ε– For small V_, b:

6. Population switching in multi-level dots: is the there room for abrupt (first order) transitions? what determines the transition width for moderate interactions? Charge sensing Qubit dephasing A basic (“trivial”) example of criticality Connecting limits of different models (Non-) Interacting resonant level versusanisotropic Anderson Full weak-to-strong coupling crossover “Applied” “Fundamental” Motivation

7. Model Hamiltonian Strongly anisotropic Anderson model, with local, tilted Zeeman field(b,ε+–ε–) V– =0  only “+” band   interacting resonant level Caution: definitions of εσ and δU here are different form those in the paper

8. Weaponry • Analytical mapping to anisotropic Kondo model via bosonisation • Pertrubative RG (in tunneling, not U!) of Yuval-Anderson-Hamann’70 • Numerical Renormalization Group • FunctionalRG Fight problems, not people!

9. Fermi liquid(Kondo) FP D << Ω D >> Ω Strategy – renormalization • Disconnected system at ε–=ε* is RG-invariant afixed point! • Tunneling is a relevant perturbation  FP is repulsive  the system is critical validity range of perturbative RG Line of critical FP!

10. Reduced to Ω Crossover to strong coupling when ~ 1 Started from Γ+ RG recipe for critical exponents • Linearize RG equations around the FP: Bosonization-based mapping: Starting (bare) value

11. Compare to numerics (alpha) • Numerics done for ε*=0 Consistent with presudo-spin Kondo regime VK,Schiller,Entin,Aharony ’07Silvestrov,Imry’07

12. Compare to numerics (beta)

13. Compare to numerics (both!) A scaling law Thanks to Amnon Aharony!

14. Some open questions • How does finite voltagedephase/modify the power-laws? • Will direct measuring of <n-> (e.g., via charge sensing) be destructive for the effect? • What if both fermionic & bosonicenvironment are present? Scaling arguments?

15. Thank you!